#linear density

Doppler Effect

A transverse wave on a string has a tension of 180 N and a linear mass density of 0.03 kg/m. What is the wavelength if t

**Relative motion** changes effective wavelength encountered. For moving source approaching stationary observer, wavelength ahead λ' = (v - v_s)/f, so f' = v/λ' = f·v/(v - v_s) > f, basis for calculating apparent pitch shift in sound. Speed: v = √((T/μ)) = √((180/0.03)) = √(6000) ≈ 77.46 m/s . Wavelength: λ = (v/v) = (77.46/20) ≈ 3.87 m . Using v = fλ and standing-wave condition fₙ = n v/(2L) or v/(4L) as applicable, calculation yields 3.87 m, illustrating frequency-length-speed interdependence and quantization by boundaries.

Ref: NCERT > Physics Book > Waves > Doppler Effect

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Wave Speed, Energy and Power

A transverse wave travels on a string with a tension of 200 N and linear mass density of 0.025 kg/m. What is the wavelen

**Energy transport** in waves scales with amplitude squared A² and frequency squared ω². Wave speed determines propagation rate, and understanding T and μ allows quantitative prediction of v and associated frequencies. Speed: v = √((T/μ)) = √((200/0.025)) = √(8000) ≈ 89.4 m/s . Wavelength: λ = (v/v) = (89.4/40) ≈ 2.24 m . Using v = fλ and standing-wave condition fₙ = n v/(2L) or v/(4L) as applicable, calculation yields 2.24 m, illustrating frequency-length-speed interdependence and quantization by boundaries.

Ref: NCERT > Physics Book > Waves > Wave Speed, Energy and Power

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Wave Speed, Energy and Power

A string of length 1.2 m and mass 0.024 kg has a wave speed of 50 m/s. What is the tension?

**Transverse wave velocity** depends on medium not frequency alone. For string under tension, v ∝ √(T/μ), calculation requires μ from mass and length, then square root evaluation, giving v in m/s, then f = v/λ for given wavelength. μ = (0.024/1.2) = 0.02 kg/m . v = √((T/μ)) ⇒ 50 = √((T/0.02)) ⇒ 50² = (T/0.02) . T = 2500 × 0.02 = 50 N . Using v = fλ and standing-wave condition fₙ = n v/(2L) or v/(4L) as applicable, calculation yields 50 N, illustrating frequency-length-speed interdependence and quantization by boundaries.

Ref: NCERT > Physics Book > Waves > Wave Speed, Energy and Power

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Transverse and Longitudinal Waves

A string of mass 0.05 kg and length 5 m is under tension. If the speed of a transverse wave is 100 m/s, what is the tens

**Mechanical waves** require medium and can be transverse or longitudinal. Sound in air is longitudinal since fluids support only compressional motion, while string waves are transverse. Progressive waves transfer energy without net mass transport, distinguishing from standing waves. μ = (mass/length) = (0.05/5) = 0.01 kg/m . v = √((T/μ)) ⇒ 100 = √((T/0.01)) . 100² = (T/0.01) ⇒ T = 10000 × 0.01 = 100 N . Using v = fλ and standing-wave condition fₙ = n v/(2L) or v/(4L) as applicable, calculation yields 100 N, illustrating frequency-length-speed interdependence and quantization by boundaries.

Ref: NCERT > Physics Book > Waves > Transverse and Longitudinal Waves

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Continuous Charge Distribution

An infinite line charge has a linear charge density \( \lambda = 2 \times 10^{-6} \, \text{C/m} \). What is the electric

**Line charge concept** extends point charge to infinite wire where symmetry dictates radial field proportional to λ and inversely proportional to distance r. λ = q/L for uniform case, field direction depends on sign of λ, outward for positive. E = (λ/2 π ε₀ r) , k = (1/4 π ε₀) = 9 × 10⁹ , so (1/ε₀) = 4 π × 9 × 10⁹ . E = (2 × 10⁻⁶/2 π × 8.854 × 10⁻¹² × 0.05) = 7.19 × 10⁵ N/C . Or, E = (2kλ/r) = (2 × 9 × 10⁹ × 2 × 10⁻⁶/0.05) = 7.2 × 10⁵ N/C . Substituting

Ref: NCERT > Physics Book > Electric Charges and Fields > Continuous Charge Distribution

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